Synchrotron X-ray diffraction study of liquid surfaces

Nuclear Instruments and Methods 208 (1983) 545-548 North-Holland Publishing Company

SYNCHROTRON

X-RAY DIFFRACTION

545

STUDY OF LIQUID

SURFACES

J. A L S - N I E L S E N Riso National Laboratory, Riscj, DK-4000 Roskilde, Denmark

P.S. P E R S H A N Division of Applied Science, Harvard University, Cambridge, Mass. 02138, USA

A spectrometer for X-ray diffraction and refraction studies of horizontal, free surfaces of liquids is described. As an illustration smectic-A layering at the surface of a liquid crystal is presented.

I. Introduction In this paper we describe an X-ray diffraction study of the surface structure of a liquid carried out at the synchrotron radiation laboratory H A S Y L A B at D E S Y in Hamburg. The instrument and the method is rather general but is probably best understood by the illustrative example provided by our study of a liquid crystal material. We shall first give a qualitative description of the method and of the spectrometer. Then follows a brief discussion of the phases of liquid crystal materials, providing the background for understanding the examples of quantitative data on the correlation of molecules normal and parallel to the liquid surface. Finally we mention other potential applications of the method.

2. Principles of method A special diffraction geometry is necessitated by the requirement that the fluid sample is being kept horizontal. In fig. 1 a monochromatic beam is extracted from the horizontal, polychromatic synchrotron beam by Bragg reflection from a perfect single crystal monochro-

.h

v

s2 Liq. X tal

t~

Fig. I. Geometry for diffraction from a horizontal surface. 0167-5087/83/0000-0000/$03.00 © 1983 North-Holland

mator M using Ge (111) or Si (111) reflections. The monochromatic beam of wavector k I is bent downwards by tilting the monochromator an angle t 1 and the height of the liquid crystal surface is adjusted so the beam always hits the same part. In our experiments the angle u is typically less than 1/40 rad, so the slit height of S1 must be quite small, say 0.02-0.10 mm. The diffracted, or refracted, beam hits an analyzer system symmetric with the monochromator. Thus only scattered photons with wavector k 2 in the vertical plane through kt will pass the analyzer crystal in its symmetric configuration and leave the analyzer horizontally when the analyzer tilt angle t 2 equals t 1. A slit S 2 at the same height as $1 puts a further restriction on the scattered photon wavevector. This slit may be real or virtually provided by a vertical position sensitive detector so that a certain channel in the PSD spectrum corresponds to the slit S2. Obviously a tilt scan with t I = t 2 and with the height h adjusted to follow the monochromatic beam provides a wavevector transfer Q~ perpendicular to the surface and probes correspondingly the structure normal to the surface with a resolution determined by the slits S 1 and S z and the synchrotron source height. The analyzer system is pivoted on a vertical axis through the sample so a wavevector transfer parallel to the surface can be obtained by turning the whole analyzer system about this axis. The relative resolution 8 Q n / Q t is here determined by the combined Darwin width W D of the M and A crystals, 8 Q n / Q t = ( k / Q t ) W D = ( 1 / 2 u ) W D. A much better resolution in an equivalent direction but within the k 1 - k 2 plane is obtained by a certain asymmetric tilt/analyzer rotation scan as will be derived explicitly in the appendix. This scan corresponds to a conventional rocking curve of a crystal where the crystal is rotated with fixed incident and scattered directions, but in the present case the sample cannot be rotated as it is fluid, so the incident and scattered directions has to be scanned with fixed sample "crystal". VII. SCATTERING/DIFFRACTION/RELATED TECHN.

546

J. Als- Nielsen, PS. Pershan / Liquid surfaces

3. Free surface and liquid crystal phases Liquid crystals consist of long molecules, typically with 2 - 3 benzene rings in the centre and a h y d r o c a r b o n tail in each end [1]. In the present compound, 80CB, the molecule has a cyanopolar head followed by 2 benzene rings, an oxygen atom with a tail of 7 CH 2 groups and terminated by C H 3. Two molecules with adjacent polar heads then constitute the symmetric building block unit [2]. In the high temperature ( T > 80°C) isotropic phase orientation as well as position of the moleular rods are random. In the nematic phase (67.51°C < T < 80°C) a spontaneous orientation of the rods has taken place but the positions are still disordered. In the smectic A-phase ( T < 67.51°C) the rods arrange themselves perpendicular to layers with a well-defined repetition distance between layers but with liquid like disorder within each layer• It is this layer formation that can be studied by X-ray diffraction. A macroscopic d o m a i n with a well defined layer-normal orientation can be obtained by putting the liquid crystal material on a glass plate treated with a surfactant to provide homeotropic alignment of the molecules at the glass-liquid crystal interface [3]. A defect (dislocation) in the smectic A layer structure will be expelled by such a rigid b o u n d a r y [4]. Furthermore, if surface tension can be neglected, such a mobile defect will be attracted towards the free surface [4]. In this sample geometry there is thus a strong tendency to form a perfect layered structure, and indeed in our initial investigation [5] we found that smectic layers with a lateral extension beyond of the order of 104 ,~ were formed at the surface of the bulk nematic phase with a penetration d e p t h into the bulk diverging as the N to A transition temperature was approached. The u n a m b i g u o u s signature of surface effects derives from the narrow, resolution limited, widths at all temperatures of scans in which the wavector transfer normal to the surface is held fixed.

4. Experimental data

80CB 67.56°C 0 0 = . 2 0 0 8 X °~

600

+,+ +

H

z

+

400

W

+

+

Z H

~2/(~00)

+

20O

+

+

+

--.99 LONGT.

1.00

Fig. 2. Longitudinal scan in the nematic phase 0.05°C above the N - A transition temperature. The width of the peak determines the penetration depth into the bulk of smectic A layering formed at the surface. The corresponding known width of bulk critical fluctuations is shown as 2/(~Q0 ).

esting and general physical effect shows up in the Fresnel fall-off. The real electron density profile at the surface is not discontinuous at the surface as assumed in s t a n d a r d electromagnetic wave reflection calculations but is a smooth curve with a characteristic length somewhat less than one molecular length. This gives rise to a form-factor effect on the Fresnel fall-off, which measures the electron density profile at the interface be....

I ....

I ....

80CB 67.56°C Q 0 = . 2 0 0 8 ~ -1

t-H U) Z LU

:11.

!

Z b~

":-.

.:.

".,

O

Our initial experiment motivated a more detailed line shape study of longitudinal scans [5]. A n example is shown in fig. 2. The k n o w n bulk correlation range ~ for smectic A fluctuations in the nematic phase correlates with the observed width as discussed in ref. 5. Data for a n extended intensity and wavevector range using a logarithmic intensity scale is shown in fig• 3• Several interesting features should be noted. Total reflection is observed for Q/Qo <0.1 with a sharp kink at the calculated critical angle corresponding to Q/Qo = 0.10. The Fresnel fall-off for Q/Qo > 0.1 including the effect of absorption fit the data perfectly up to say Q/Qo ~- 0.2. For wavevectors longer than Q/Qo = 0.2 another inter-

1.01

WAVEVECTOR Q / 0 0

:!

{.9 O %

.5 LONGT.

..

1 0 WAVEVECTOR 0/C~0

Fig. 3. Total reflection ( Q / Q 0 < 0.1), Fresnel partial reflection and diffraction from the smectic A layering at the surface.

J. AIs-Nielsen, P.S. Pershan / Liquid surfaces

2000

80CB 6 7 . 3 7 *C

547

tional Laboratory, b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t No. DRM-79-19479, a n d by the Joint Services Electronics Program (U.S. Army, Navy a n d Air Force) u n d e r G r a n t No. 14-75-C-0648.

+

+ + +

~000

+

+ ±

Appendix

+

++

++++ J t

+

+++. i

TRANSV.

J

[

+

/

I

5

4 .1o- , O WAVEVECTOFI

Fig. 4. "Rocking-curve" of the (0, 0, 1) peak in the smectic A phase, c.f. appendix eqs. (5) and (6).

tween liquid a n d gas [6,7]. The sharp decrease a r o u n d Q/Qo = 1.05 is due to destructive interference between Fresnel diffracted a n d layer diffracted waves. A detailed analysis including model fitting to the data will be published elsewhere. In fig. 4 we give an example of " r o c k i n g curve" data as explained in the end of section 2. The liquid crystal is here in the smectic A phase. It is quite r e m a r k a b l e to o b t a i n a rocking curve width of only 0.015 ° fwhm on a sample, which after all is fluid. In ,h,-1 the half-width h a l f - m a x i m u m is 2.5 × 10 -5, that is the lateral correlation range of these layers is b e y o n d 40.000 or 4 # m , an astonishing number, when it is b o r n e in m i n d that it is d e t e r m i n e d by X-ray diffraction.

5.

In this appendix we derive explicitly the relation between spectrometer setting a n d wavevector transfer a n d discuss scans in symmetry directions. The coordin a t e system and definitions of angles are given in fig. 5. The coordinates of vectors in reciprocal space is given in table 1 with the following comments: 1st row: The m o n o c h r o m a t o r is tilted the angle t l a r o u n d the y-axis. 2 n d row: T h e wavevector k 0 selected from the synchrotron b e a m is horizontal a n d 0 is the Bragg angle when t I = 0. 0 is kept at a fixed value. 3rd row: Entries follow from the Bragg condition k 1 = ~1 + k0. 4th row: The analyzer crystal is rotated the angle t 2 a r o u n d the y-axis. A horizontal line through the reflecting planes is turned an angle r 2 with respect to the y-axis. 5th row: The wavevector k 3 after the analyzer crystal makes the angle q~ with the horizontal plane a n d angle a with the x-axis, a is close to, but not necessarily equal to, 0. 6th row: Entries follow from k 2 = k 3 - 'r2. 7th row: Entries follow from Q = k 2 - k 1. Since the scattering is elastic Ik i r2 = ]k 2 p= k 2 leading to the following relations: k = ~'/(2 sin 0 cos t , ) ,

(1)

Conclusions

sin a cos t I + sin q, sin t 2

sin(a-r:) = The discussion of the diffraction data in the previous section leads naturally to investigating a liquid crystal material with one or more heavy atoms like iodine in the molecule in order to alter the electron density profile drastically. This m e t h o d of investigating the liquidsurface electron density profile m a y turn out to be i m p o r t a n t in general. In-plane correlations m a y be studied by a liquid crystal material exhibiting the smectic B-phase. Precursor effects at the surface of the smectic B ordering m a y be very interesting to study. Again the m e t h o d m a y be carried over to other materials or interfaces such as, for instance, surfactant layers on water

cos q~ cos 12

(2)

The condition set by the slit S z relates u, v a n d q~ by: l 0 tan v + l I tan q~ = l 0 tan u.

Side view

(3)

I,

ko

[8]. The excellent research conditions provided by H A S Y L A B a n d the c o m p e t e n t assistance of Risq technical staff m e m b e r s E. D a h l Petersen, S. Bang, J. Linderholm, a n d J. M u n c k are gratefully acknowledged. This work was s u p p o r t e d in part b y grants from the D a n i s h N a t i o n a l Science F o u n d a t i o n , by the Ris6 Na-

Top view

Fig. 5. Coordinate system and definitions of angles. VII. SCATTERING/DIFFRACTION/RELATED TECHNIQUES

548

J. Als-Nielsen, P.S. Pershan / Liquid surfaces

Table 1 Vectors in reciprocal space Quantity

Notation

Components

Monochromator rec. lat. vector Synchrotron wavevector Incident wavevector Analyzer rec. lat. vector Detector w a v e c t o r Scattered wavevector

~1 ko k1 ~2 k3 k2

Wavevector transfer

Q

Expansion from Bragg point

8Q/(r/2)

~=rcostl,~=O,~= "r sin t I = -ksin0.~=kcos0,~.=0 .?='rcost I-ksin0,y=kcos0.g.-~sint 1 ~= "r cos t 2 cos r 2 , .i' = "r cos t 2 sin r2, d = ~" sin t 2 - -ksinacos~,~=k c o s c ~ c o s q ~ , d - k sin = - k sin ct cos ,~ + "r cos l 2 cos r 2, f = k cos c~ cos ~ + "r cos t 2 sin r2, d=k sin~+~'sint 2 - k(sin 0 - s i n a cos q~)-'r(cos t I - c o s t 2 cos rx) .9=k( cosO+cosc~cos~)+'ccost2sinr 2 d = k s i n , ~ + ' r ( s i n t 2 + s i n t I) = 2 sin to(3t ~- 3 t 2 ) - ( c o t 0 / c o s t~)3c~ .9 = 2 cos toar 2 - 3 a / c o s t o 2 = 2 cos t0(3t 1 + 3t2)+ 8~,b/(cos t o sin 0)

L o n g i t u d i n a l w a v e v e c t o r transfer

6a=totanO(dt2-atl)+(to/cosO)a4,+ar

F r o m s y m m e t r y it is i m m e d i a t e l y a p p a r e n t t h a t a p u r e l y l o n g i t u d i n a l w a v e v e c t o r t r a n s f e r is o b t a i n e d for t I = t 2, r 2 = 0. Since u = v eq. (3) i m p l i e s 4, = 0 a n d f r o m eq. (2) follows a = 0. By i n s p e c t i o n o f t h e table we find i n d e e d Qx = Q ~. = 0 a n d Q= = 2~'sin t~. It s h o u l d be n o t e d t h a t t h e l o n g i t u d i n a l w a v e v e c t o r t r a n s f e r is independent o f wavelength. E x p a n s i o n a r o u n d (0, O, Qo)

L e t us e x a m i n e t h e g e o m e t r y for w a v e v e c t o r t r a n s fers in t h e vicinity of (0, 0, Q0 = 2~-sin to) w i t h t h e p a r t i c u l a r a i m o f d e r i v i n g t h e s p e c t r o m e t e r s e t t i n g for a p u r e l y t r a n s v e r s e w a v e v e c t o r d e v i a t i o n in t h e k j - k : p l a n e . T h i s is useful for d e t e r m i n i n g for e x a m p l e t h e m o s a i c i t y o f a l a y e r e d s t r u c t u r e as it o c c u r s in t h e s m e c t i c A p h a s e o f liquid crystals. W e shall u s e t h e f o l l o w i n g n o t a t i o n in a linear e x p a n s i o n : c~ = O + 6 a , t~ = t o + 6t~,

4, = 64,,

(2a)

x.

F i n a l l y t h e r e q u i r e m e n t of p u r e l y transverse 6 Q tog e t h e r with eq. (3a) l e a d s to: llat z = - (2l 0 + l,)6t,.

(5)

E l i m i n a t i o n of 8 3 a n d 64, finally yields: [1 + 2 s i n : 0 / c o s 2 0 1 6 r : = 2 t 0 ( l 0 + l , ) ×{l,[tan20+(l~-lo)/(l

~+lo)

tan0]atl}

i (6)

In s u m m a r y t h e n a p u r e l y t r a n s v e r s e s c a n in the k I - k 2 p l a n e is o b t a i n e d b y tilting t h e a n a l y z e r in t h e o p p o s i t e d i r e c t i o n of t h e m o n o c h r o m a t o r in t h e ratio of ( 2 l 0 + l~) : l~ a n d at the s a m e t i m e r o t a t i n g the a n a l y z e r a r o u n d a vertical axis in t h e ratio g i v e n by eq. (6). In the d a t a p r e s e n t e d in figs. 2 - 4 t h e p a r a m e t e r s were: 0=

13.642 ° ,

S 2= 1 × 10mm,

~= 1.924A -l, 10=575mm,

S t=0.02x2mm, 11=620mm.

rz = a r 2,

t 2 = t o + 8t 2 .

References T h e w a v e v e c t o r deviation f r o m (0, 0, Q0) is g i v e n in r o w 8 of the table. T h e r e q u i r e m e n t t h a t 6 Q is p u r e l y t r a n s v e r s e in t h e k I - k 2 plane can be expressed by the condition (k I x £ ) . 8 Q = 0. U s i n g t h e e x p a n s i o n e x p r e s s i o n for 6 Q a n d f u r t h e r m o r e utilizing t h a t t h e tilt a n g l e s are s m a l l l e a d s to: 63 = to tan 20(6t 1 - 6t2)-

(2 s i n Z 0 / c o s 2 0 ) 8 r 2.

(4)

In the linear e x p a n s i o n eqs. (3) a n d (2) b e c o m e : 64, = 2 1 o / ( l o + 1~) sin O( 6t~ - 6 t 2 ) ,

(3a)

[1] P.G. de Gennes, The physics of liquid crystals (Clarendon Press, Oxford, 1974). [2] P.E. Cladis, R.K. Bogardus, W.B. Daniels and G.N. Taylor, Phys. Rev. Lett. 39 (1977) 720. [3] F.J. Kahn, Appl. Phys. Lett. 22 (1973) 387. [4] P.S. Pershan, J. Appl. Phys. 45 (1974) 1590. [5] J. Als-Nielsen, F. Christensen and P.S. Pershan, Phys. Rev. Lett. 48 (1982) 1107. [6] L.G. Parratt, Phys. Rev. 95 (1954) 359. [7} D.H. Bilderback, Proc. SPIE 315 (1981) 30. [8] P.S. Pershan, J. d. Phys. C3 40 (1979) 423.